Geometry optimization
Fixed cell
We use DFTK and the GeometryOptimization package to find the minimal-energy bond length of the $H_2$ molecule. First we set up an appropriate DFTKCalculator (see AtomsCalculators integration), for which we use the LDA model just like in the Tutorial for silicon in combination with a pseudodojo pseudopotential (see Pseudopotentials).
using DFTK
using PseudoPotentialData
pseudopotentials = PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf")
calc = DFTKCalculator(;
model_kwargs = (; functionals=LDA(), pseudopotentials), # model_DFT keyword arguments
basis_kwargs = (; kgrid=[1, 1, 1], Ecut=10) # PlaneWaveBasis keyword arguments
)DFTKCalculator(functionals=Xc(lda_x, lda_c_pw), pseudopotentials=PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf"), Ecut=10, kgrid=[1, 1, 1])Next we set up an initial hydrogen molecule within a box of vacuum. We use the parameters of the equivalent tutorial from ABINIT and DFTK's AtomsBase integration to setup the hydrogen molecule. We employ a simulation box of 10 bohr times 10 bohr times 10 bohr and a pseudodojo pseudopotential.
using LinearAlgebra
using Unitful
using UnitfulAtomic
r0 = 1.4 # Initial bond length in Bohr
a = 10.0 # Box size in Bohr
cell_vectors = [[a, 0, 0]u"bohr", [0, a, 0]u"bohr", [0, 0, a]u"bohr"]
h2_crude = periodic_system([:H => [0, 0, 0.]u"bohr",
:H => [r0, 0, 0]u"bohr"],
cell_vectors)FlexibleSystem(H₂, periodicity = TTT):
cell_vectors : [ 10 0 0;
0 10 0;
0 0 10]u"a₀"
Atom(H, [ 0, 0, 0]u"a₀")
Atom(H, [ 1.4, 0, 0]u"a₀")
Finally we call minimize_energy! to start the geometry optimisation. We use verbosity=2 to get some insight into the minimisation. With verbosity=1 only a summarising table would be printed and with verbosity=0 (default) the minimisation would be quiet.
using GeometryOptimization
results = minimize_energy!(h2_crude, calc; tol_forces=2e-6, verbosity=2)
nothing # hiden Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.110639496168 -0.82 0.80 7.0 1.64s
2 -1.117203460545 -2.18 -1.88 0.80 1.0 887ms
3 -1.117250567098 -4.33 -2.71 0.80 1.0 11.3ms
4 -1.117251170511 -6.22 -3.58 0.80 1.0 11.0ms
5 -1.117251188157 -7.75 -4.04 0.80 1.0 11.1ms
6 -1.117251189934 -8.75 -4.60 0.80 1.0 11.1ms
7 -1.117251190132 -9.70 -5.86 0.80 1.0 42.0ms
8 -1.117251190143 -10.99 -5.94 0.80 2.0 12.2ms
9 -1.117251190143 -12.62 -6.20 0.80 1.0 11.0ms
10 -1.117251190143 -12.96 -6.83 0.80 1.0 11.1ms
11 -1.117251190143 -14.16 -7.45 0.80 1.0 136ms
12 -1.117251190143 + -15.05 -8.26 0.80 1.0 11.6ms
13 -1.117251190143 + -15.65 -9.00 0.80 1.0 11.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -9.38 0.80 1.0 1.62s
2 -1.117251190143 -15.35 -10.38 0.80 1.0 10.5ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 9.77s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517993617 -2.46 0.80 1.0 7.25ms
2 -1.117520586039 -5.59 -3.49 0.80 1.0 49.4ms
3 -1.117520613009 -7.57 -4.23 0.80 1.0 10.5ms
4 -1.117520613449 -9.36 -5.17 0.80 1.0 10.3ms
5 -1.117520613464 -10.82 -5.95 0.80 1.0 10.3ms
6 -1.117520613465 -12.18 -6.47 0.80 1.0 10.3ms
7 -1.117520613465 -13.42 -7.56 0.80 1.0 27.3ms
8 -1.117520613465 + -14.61 -8.27 0.80 1.0 10.4ms
9 -1.117520613465 -14.45 -8.62 0.80 1.0 10.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264674 -1.69 0.80 1.0 12.9ms
2 -1.118339043937 -4.04 -2.70 0.80 1.0 9.70ms
3 -1.118340011116 -6.01 -3.44 0.80 1.0 9.88ms
4 -1.118340028555 -7.76 -4.36 0.80 1.0 10.1ms
5 -1.118340029140 -9.23 -5.12 0.80 1.0 16.1ms
6 -1.118340029167 -10.57 -5.62 0.80 1.0 10.2ms
7 -1.118340029169 -11.73 -6.68 0.80 1.0 10.3ms
8 -1.118340029169 -13.55 -7.58 0.80 1.0 53.3ms
9 -1.118340029169 + -14.54 -8.12 0.80 1.0 10.8ms
10 -1.118340029169 -14.75 -8.55 0.80 1.0 286ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029169 -9.05 0.80 1.0 6.92ms
2 -1.118340029169 + -14.88 -10.33 0.80 1.0 9.91ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029169 -11.10 0.80 1.0 6.76ms
2 -1.118340029169 + -14.40 -11.49 0.80 1.0 9.78ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029169 │ │ 0.00297055 │ 1.71s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721507 -3.09 0.80 1.0 6.95ms
2 -1.118346864760 -6.84 -4.11 0.80 1.0 9.91ms
3 -1.118346866237 -8.83 -4.86 0.80 1.0 9.88ms
4 -1.118346866263 -10.59 -5.78 0.80 1.0 10.0ms
5 -1.118346866264 -12.09 -6.55 0.80 1.0 10.0ms
6 -1.118346866264 -13.39 -7.04 0.80 1.0 18.8ms
7 -1.118346866264 + -14.75 -8.13 0.80 1.0 10.6ms
8 -1.118346866264 -15.05 -8.95 0.80 1.0 10.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354724004 -2.61 0.80 1.0 6.83ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 9.90ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 9.86ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 9.93ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 17.1ms
6 -1.118356101386 -12.45 -6.55 0.80 1.0 10.4ms
7 -1.118356101386 -13.64 -7.65 0.80 1.0 10.5ms
8 -1.118356101386 -14.88 -8.45 0.80 1.0 10.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -9.24 0.80 1.0 6.77ms
2 -1.118356101386 + -15.35 -9.82 0.80 1.0 13.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 6.80ms
2 -1.118356101386 + -14.22 -11.46 0.80 1.0 9.90ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 282ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356102848 -4.94 0.80 1.0 6.98ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 9.87ms
3 -1.118356102877 -12.53 -6.70 0.80 1.0 9.79ms
4 -1.118356102877 -14.12 -7.62 0.80 1.0 10.0ms
5 -1.118356102877 + -14.88 -8.39 0.80 1.0 16.8ms
6 -1.118356102877 -14.51 -8.89 0.80 1.0 10.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356104771 -4.40 0.80 1.0 9.56ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 9.94ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 9.84ms
4 -1.118356105120 -13.21 -7.09 0.80 1.0 12.9ms
5 -1.118356105120 -15.05 -7.86 0.80 1.0 10.4ms
6 -1.118356105120 -15.35 -8.35 0.80 1.0 10.3ms
7 -1.118356105120 -15.35 -9.45 0.80 1.0 13.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -10.08 0.80 1.0 6.90ms
2 -1.118356105120 + -14.21 -11.02 0.80 1.0 9.91ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 6.84ms
2 -1.118356105120 -14.37 -12.44 0.80 1.0 12.7ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 1.00309e-8 │ 266ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
Structure after optimisation (note that the atom has wrapped around)
results.systemFlexibleSystem(H₂, periodicity = TTT):
cell_vectors : [ 10 0 0;
0 10 0;
0 0 10]u"a₀"
Atom(H, [-0.0431829, -2.09533e-34, -1.55967e-34]u"a₀")
Atom(H, [ 1.44318, -1.3127e-34, 7.22625e-35]u"a₀")
Compute final bond length:
rmin = norm(position(results.system[1]) - position(results.system[2]))
println("Optimal bond length: ", rmin)Optimal bond length: 1.486365868323205 a₀
Our result (1.486 Bohr) agrees with the equivalent tutorial from ABINIT.
Variable cell
Recent versions of GeometryOptimization support cell optimization as well by passing variablecell=true to minimize_energy!.
For a plane-wave code like DFTK, variable cells pose an additional challenge: changes to the cell's size affect the used plane waves, leading to discontinuities in the energy and stresses. This can cause the geometry optimization to struggle and/or fail to converge.
A practical strategy to overcome this problem is Energy cutoff smearing. As a demonstration let us find the optimal lattice constant of silicon.
Like before we define a calculator, this time with a kinetic_blowup set to use energy cutoff smearing:
calc = DFTKCalculator(;
model_kwargs = (; functionals=LDA(), pseudopotentials,
kinetic_blowup=BlowupCHV()),
basis_kwargs = (; kgrid=[2, 2, 2], Ecut=10)
)DFTKCalculator(functionals=Xc(lda_x, lda_c_pw), pseudopotentials=PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf"), Ecut=10, kgrid=[2, 2, 2])And here is our starting silicon structure:
a = 10.0u"bohr" # Approximate Silicon lattice constant
cell_vectors = a/2 * [[0, 1, 1], [1, 0, 1], [1, 1, 0]]
initial_silicon = periodic_system([:Si => ones(3)/8,
:Si => -ones(3)/8],
cell_vectors;
fractional=true)FlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [ 0 5 5;
5 0 5;
5 5 0]u"a₀"
Atom(Si, [ 1.25, 1.25, 1.25]u"a₀")
Atom(Si, [ -1.25, -1.25, -1.25]u"a₀")
We now minimize, passing variablecell=true:
using GeometryOptimization
results = minimize_energy!(initial_silicon, calc; variablecell=true,
tol_virial=2e-6, verbosity=2)
nothing # hiden Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.335352360974 -0.81 0.80 7.3 392ms
2 -8.340909603303 -2.26 -1.69 0.80 1.3 41.0ms
3 -8.341203711551 -3.53 -2.79 0.80 2.3 15.8ms
4 -8.341214233336 -4.98 -3.26 0.80 3.3 18.6ms
5 -8.341214269137 -7.45 -3.97 0.80 1.3 14.1ms
6 -8.341214272347 -8.49 -5.22 0.80 2.0 15.1ms
7 -8.341214272517 -9.77 -5.70 0.80 4.0 20.6ms
8 -8.341214272518 -11.87 -6.40 0.80 2.0 15.1ms
9 -8.341214272518 -13.07 -7.44 0.80 2.7 34.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -9.05 0.80 1.0 37.8ms
2 -8.341214272518 + -14.75 -10.02 0.80 1.0 12.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -10.25 0.80 1.0 11.3ms
2 -8.341214272518 + -14.75 -10.51 0.80 1.0 12.5ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 0 │ -8.341214272518 │ │ 4.86142e-33 │ 0.163458 │ -0.16 │ 17.2s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.350404694746 -1.20 0.80 8.7 74.6ms
2 -8.351871327008 -2.83 -1.88 0.80 2.0 16.4ms
3 -8.351926123941 -4.26 -2.79 0.80 3.0 18.1ms
4 -8.351928511201 -5.62 -3.86 0.80 3.3 19.6ms
5 -8.351928514542 -8.48 -5.17 0.80 3.0 18.1ms
6 -8.351928514651 -9.97 -5.93 0.80 4.0 21.4ms
7 -8.351928514651 -12.19 -7.11 0.80 3.7 19.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353402075518 -1.52 0.80 8.0 31.4ms
2 -8.353732391083 -3.48 -2.21 0.80 1.0 13.3ms
3 -8.353757270372 -4.60 -2.99 0.80 3.0 18.5ms
4 -8.353757526836 -6.59 -4.25 0.80 2.3 39.5ms
5 -8.353757531711 -8.31 -5.36 0.80 3.0 19.7ms
6 -8.353757531722 -10.97 -6.35 0.80 3.3 19.3ms
7 -8.353757531722 -13.00 -7.26 0.80 3.0 20.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531722 -8.17 0.80 1.0 11.8ms
2 -8.353757531722 -14.75 -9.21 0.80 1.0 13.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531722 -9.73 0.80 1.0 11.8ms
2 -8.353757531722 + -14.27 -10.08 0.80 1.0 18.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531722 -10.32 0.80 1.0 11.9ms
2 -8.353757531722 + -14.27 -10.67 0.80 1.0 13.3ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 1 │ -8.353757531722 │ │ │ 0.0156413 │ -0.016 │ 3.14s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353642347329 -3.06 0.80 8.0 31.5ms
2 -8.353746570395 -3.98 -3.29 0.80 1.0 13.7ms
3 -8.353770534550 -4.62 -3.58 0.80 4.7 24.5ms
4 -8.353770534594 -10.35 -3.84 0.80 1.0 13.7ms
5 -8.353770536958 -8.63 -4.43 0.80 1.0 26.5ms
6 -8.353770537300 -9.47 -4.70 0.80 1.3 14.8ms
7 -8.353770537363 -10.20 -4.89 0.80 2.0 16.0ms
8 -8.353770537398 -10.46 -5.88 0.80 1.0 13.8ms
9 -8.353770537399 -11.83 -6.70 0.80 3.3 20.3ms
10 -8.353770537399 -13.75 -6.80 0.80 2.7 18.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353788835860 -2.00 0.80 8.0 31.8ms
2 -8.353880576819 -4.04 -2.71 0.80 1.0 13.6ms
3 -8.353895063247 -4.84 -3.36 0.80 3.7 25.3ms
4 -8.353895107578 -7.35 -4.30 0.80 2.0 16.4ms
5 -8.353895110736 -8.50 -5.07 0.80 3.7 20.6ms
6 -8.353895110778 -10.38 -5.82 0.80 2.7 18.1ms
7 -8.353895110779 -11.77 -6.76 0.80 2.7 22.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110779 -7.50 0.80 1.0 12.2ms
2 -8.353895110779 -14.45 -8.35 0.80 1.0 18.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110779 -8.82 0.80 1.0 12.2ms
2 -8.353895110779 -14.27 -9.02 0.80 1.0 13.6ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110779 -9.33 0.80 1.0 12.2ms
2 -8.353895110779 + -14.75 -9.53 0.80 1.0 13.5ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 2 │ -8.353895110779 │ -3.86 │ 1.07896e-32 │ 0.000646168 │ -0.00065 │ 1.67s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353823546659 -3.60 0.80 7.7 36.7ms
2 -8.353882947632 -4.23 -3.85 0.80 1.0 13.9ms
3 -8.353895212184 -4.91 -4.03 0.80 5.3 26.9ms
4 -8.353895212315 -9.88 -4.26 0.80 1.0 13.7ms
5 -8.353895212473 -9.80 -4.76 0.80 1.0 17.1ms
6 -8.353895212508 -10.46 -5.01 0.80 1.0 13.6ms
7 -8.353895212521 -10.88 -6.30 0.80 1.7 14.8ms
8 -8.353895212522 -11.96 -6.52 0.80 3.3 22.9ms
9 -8.353895212522 -13.91 -6.70 0.80 1.0 13.5ms
10 -8.353895212522 -14.27 -7.21 0.80 1.0 13.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353799507279 -3.34 0.80 7.7 33.6ms
2 -8.353877625165 -4.11 -3.73 0.80 1.0 13.0ms
3 -8.353895360921 -4.75 -4.03 0.80 5.3 24.9ms
4 -8.353895362545 -8.79 -4.63 0.80 1.3 17.1ms
5 -8.353895362667 -9.91 -5.13 0.80 2.0 15.5ms
6 -8.353895362684 -10.77 -5.75 0.80 2.0 16.0ms
7 -8.353895362686 -11.88 -6.52 0.80 2.3 16.7ms
8 -8.353895362686 -13.30 -7.22 0.80 3.0 22.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -8.02 0.80 1.0 11.4ms
2 -8.353895362686 -14.45 -9.13 0.80 1.0 12.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.46 0.80 1.0 11.4ms
2 -8.353895362686 + -Inf -9.67 0.80 1.0 12.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.85 0.80 1.0 11.5ms
2 -8.353895362686 + -14.75 -10.23 0.80 1.0 12.8ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────────┤
│ 3 │ -8.353895362686 │ -6.60 │ │ 1.71267e-6 │ -1.7e-6 │ 1.50s │
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────────┘
Structure after optimization
results.systemFlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [-4.2362e-51 5.27783 5.27783;
5.27783 -4.51861e-51 5.27783;
5.27783 5.27783 -4.51861e-51]u"a₀"
Atom(Si, [ 1.31946, 1.31946, 1.31946]u"a₀")
Atom(Si, [-1.31946, -1.31946, -1.31946]u"a₀")
Since here the cell was rescaled but its shape did not change, we directly extract the optimal lattice constant from one of the cell vectors:
using AtomsBase
amin = AtomsBase.cell_vectors(results.system)[1][2]*2
println("Optimal lattice constant: ", amin)Optimal lattice constant: 10.55566293659422 a₀
Note that while for silicon the positions of the atoms are fixed by symmetry, in general a variable cell optimization will try to optimize both the cell and the positions of the individual atoms.